# Exponentials, Logarithms & the Natural Log

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• 0:06 What is an Exponential?
• 1:05 Rules of Natural Log
• 2:43 Carbon Dating with Natural Log
• 5:57 More Rules of Logarithms
• 7:21 Lesson Summary

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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Use the properties of exponentials and logarithms to learn how carbon dating works. This lesson covers properties of a natural log and rules of logarithms.

## What is an Exponential?

Let's return to a scientist studying an old math book. He wants to find out how old it is using carbon dating. He knows that he needs to use an exponential to do this. He has y = a^x, where a is a base and x is the exponent. You can remember all you need to know about powers and exponentials by thinking back to the bunny problem. You start out with two bunnies, then they multiply. First you have 2, then you have 2 * 2 ... * 2 * 2 * 2. You have 2^x bunnies, where you're multiplying 2 times itself x times. In the case of the bunnies, 2 was our base.

But there's a special type of base that we see everywhere in the physical world. That base is e. e is just a constant, like pi, except that e is Euler's number, and it's roughly 2.72.

## Rules of Natural Log

Now, e follows all of the properties of other powers, like (e^x)(e^y) = e^(x + y). e^x / e^y = e^(x - y). e^0 = 1. (e^n)^m = e^(nm). In all of these cases, e is just 2.72. When I say e^0=1, what I'm saying is that 2.72 ... ^0 = 1.

What's special about e^x is its inverse. f(x) = e^x, but the inverse of f(x) is the natural log of x: ln(x). So f^-1(f(x)) = ln(e^x) = x. Let's go through some properties of e and the natural log. We know that e^(ln(x)) must be x. We know that ln(e^x) = x. But we also know that ln(x/y) = ln(x) - ln(y). Similarly, ln(xy) = ln(x) + ln(y). ln(x^n) = nln(x).

## Carbon Dating Using Natural Log

So let's use these properties to help date an old math book. Let's use carbon dating. The half-life of a particular type of carbon is 5,730 years. Carbon dating is done by setting the percentage of this special type of carbon as being equal to e to some constant number times t (e^(Ct)).

If you set the percentage of carbon that you have, and you know what this C is, you can calculate how old your item is. First, we need to find out what this number C is. I said that the half-life of carbon is 5,730 years. That means that after 5,730 years, we have 50% of our carbon left. So 50% carbon = e^(C(5730)). Okay, well, 50% carbon is the same as .5, so I have the equation .5 = e^(5730C). To solve for C, I need to first take the natural log of both the left-hand side and the right-hand side. I get ln(.5) = ln(e^(5730C)). If I use my property that ln(e^x) = x, the right-hand side is equal to 5730C. If I divide both the left and right side by 5730, I can calculate C. C = ln(.5)/5730, which is about equal to -0.000121. So let's put that back into our original equation. The percent carbon that we have is equal to e^(-0.000121t). So if I plug in the percent carbon that I have on the left-hand side, I can solve this equation for the age of my material.

Say that there's 30% of this special carbon in our math book. I'm going to take the natural log of both the left-hand side and the right-hand side. I get ln(.3) = ln(e^(-0.000121t)). I know that ln(e^x) = x, so the right-hand side simplifies to -0.000121t. I can divide both sides by that number, and I find that t = (ln(.3))/-0.000121, which is about 9,950 years old. That's a really old math book. While we see e and the natural log a lot in the physical world, there are other bases that have similar properties.

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